Multiplicity and asymptotic behavior of normalized solutions to p-Kirchhoff equations

In this paper, we study a type of p-Kirchhoff equation $$-(a+b{\int }_{{\mathbb{R}}^3}{\mid \mathrm{\nabla }u\mid }^{p}dx){\Delta }_{p}u=\lambda {\mid u\mid }^{p-2}u+{\mid u\mid }^{q-2}u,x\in {\mathbb{R}}^3$$ with the prescribed mass $${\left({\int }_{{\mathbb{R}}^3}{\mid u\mid }^{p}dx\right)}^{\frac{1}{p}}=c>0$$ where $a>0,b>0,\frac{3}{2}<p<3,p<q<p^{\ast }:=\frac{3p}{3-p}$,${\Delta }_{p}u=div\left({\mid \mathrm{\nabla }u\mid }^{p-2}\mathrm{\nabla }u\right)$ is the $p$-Laplacian of $u$, $\lambda \in \mathbb{R}$ is Lagrange multiplier. We consider both $L^p$-subcritical , $L^p$-critical and $L^p$-supercritical cases. Precisely, in the $L^p$-subcritical and $L^p$-critical cases, we obtain the existence and nonexistence of the normalized solutions for the $p$-Kirchhoff equation. In the $L^p$-supercritical case, we obtain the existence of radial ground sates and multiplicity of radial normalized solutions for the $p$-Kirchhoff equation. Furthermore, we study the asymptotic behavior of normalized solutions when $b\to 0^{+}$. Besides, when $\frac{3}{2}<p\le 2$, benefit from the uniqueness(up to translations) of optimizer for Gargliardo-Nirenberg inequality, we show the existence and uniqueness of normalized solutions and provide the accurate descriptions.